\(\mathbf {Example\;2.5}\) : These four generators form what is actually
known as the NC \(4-\) Torus [1]}, that is, the smooth
subalgebra \(C^{\infty }(\mathbb {T}_{\Theta }^{4})\)  of the universal
\(C^{*}-\) algebra \(A_{\Theta }\)  defined by
\(A_{\Theta }\ni a\doteq \sum _{r\in \mathbb {Z}^{4}}a_{r}u^{r},\) 
\(u^{r}\doteq e^{\pi i\sum _{\underset{j<k}{j,k=1}}^{4}r_{j}\Theta _{jk}r_{k}}u_{1}^{r_{1}}u_{2}^{r_{2}}u_{3}^{r_{3}}u_{4}^{r_{4}},\) 
\(\Theta \doteq \left(\begin{array}{cccc}0 & \theta (\overline{P}_{N_{1}},P_{N_{2}}) & \theta (\overline{P}_{N_{1}},\overline{P}_{N_{2}}) & \theta (\overline{P}_{N_{1}},P_{N_{3}})\\-\theta (\overline{P}_{N_{1}},P_{N_{2}}) & 0 & \theta (P_{N_{2}},\overline{P}_{N_{2}}) & \theta (P_{N_{2}},P_{N_{3}})\\-\theta (\overline{P}_{N_{1}},\overline{P}_{N_{2}}) & -\theta (P_{N_{2}},\overline{P}_{N_{2}}) & 0 & \theta (\overline{P}_{N_{2}},P_{N_{3}})\\-\theta (\overline{P}_{N_{1}},P_{N_{3}}) & -\theta (P_{N_{2}},P_{N_{3}}) & -\theta (\overline{P}_{N_{2}},P_{N_{3}}) & 0\end{array}\right).\) 
